Hi Chris,
I gave some further thought to this question after posting. Of the three ships, from the photos that are available online, it appears that the bridge front of the Manchester Port was wood faced. The Progress and City were white painted metal. The quotes that state the City was first a mine-layer from the outbreak of war and then a naval support vessel in the Far East, appear inconsistent with the OP's grandfather's recollections of his Northern waters voyages on the ML vessel. Whilst far from conclusive, it suggests that the picture provided might be of the Progress rather than the City.
Regards,
David